On the differential geometric Schottky problem

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We consider the action of the complex orthogonal group on a simply connected minimal surface in the n-dimensional Euclidean space by using the Weierstrass representation theorem and decide the minimal surface appearing as limits of this action. An appearing minimal surface is a generalization of the Enneper minimal surface. If the given minimal surface is not holomorphic, then this action makes the minimal surface to be unstable. The reason is that the bifurcation (Catastrophe phenomena) occurs.We consider the Plateau problem for the frame (a curve in the 3 dimensional Euclidean space) like the Nitsch frame. We study the bifurcation of a minimal surface by a deformation of the frame and see the geometric visualization of the catastrophe phenomenon by a computer simulation.We consider a deformation of a minimal surface in a torus by a deformation of the torus. This is deeply relative to the Differential geometric Schottky problem (investigate the curvedness of the space of Riemann matrices in the Siegel upper half space). We consider the catastrophe set in the cotangent bundle of the space of the deformation of tori. Then we obtain a Lagrangian cone with a complex structure and generalize the construction.We consider a minimal surface system in the Euclidean space with higher co-dimension for the Dirichlet problem (a given boundary map). For the boundary map, there exists a solution and no solution.. We may consider that the solution with singularity appears under the deformation of the boundary map. In fact, Lawson and Osserman obtain a solution as a minimal cone. We generalize their construction and give many solutions as minimal cones. Moreover we see that a minimal cone have the deformation space.Hence it is important to study the deformation of minimal surfaces which contains our results.